Outline

1. Aggregation-Fragmentation Modelsfor Protein PolymerizationMarie DoumicJournées non linéaires – Journées VenakidesInstitut de Mathématiques de JussieuJune 24th, 2010 with: V. Calvez, P. Gabriel, T. Goudon, N. Lenuzza, T. Lepoutre, B. Perthame and biologists: F. Mouthon, H. Rezaei Marie DOUMIC Journées Venakides

2. Marie DOUMIC Journées Venakides Outline A (very) little bit about the biology of Prion and Alzheimer Review on existing mathematical models for prion disease & previous results Some recent results: Link between discrete and continuous modelling Eigenvalue problem Parameters dependency Perspectives

3. Marie DOUMIC Journées Venakides • What is a common point of • Prion (Madcow, Kuru, Creutzfeldt-Jakob disease) • Alzheimer • Huntington • … and some others: • Neurodegenerative diseases • characterized by abnormal accumulation of • insoluble fibrous protein aggregates • called AMYLOIDS

4. Where do these fibrils come from ? Healthy state: the considered protein is present (APP for Alzheimer, PrP for Prion, HTT for Huntington…) BUT is monomeric (N.B.: mono=1, poly = several) Disease : Misfolding of the protein, leading to its polymerization onto long fibrils (monomers attach to one another) Specificity of Prion disease: transmissible PRION: PRoteinasceous Infectious Only (Prusiner 1982, Griffith 1967, Nobel Prize)

5. Marie DOUMIC - BONN October, 6th, 2009 i-mer (i+1)-mer Polymerization monomer + i+1

6. i-mer j-mer (i-j)-mer Fragmentation + j monomers if j < i0

7. i-mer j-mer (i+j)-mer Coagulation +

8. i-mer monomer (i-1)-mer + Depolymerization

9. Discrete Prion model Masel, Jansen, Nowak, 1999: synthesis fragmentation Monomer density degradation i-polymer density polymerization Size of a nucleus = minimal size of stable polymers

10. Remarks on discrete polymerization models In Biology and Biophysics • Huge literature • Simulation tools e.g. softwares “PREDICI”, “POLYRED” (e.g. see articles by Wulkow et al., Choi et al.) • Main difficulty: large (infinite…) number of parameters often drastic simplifying assumptions In Mathematics • Family of coagulation-fragmentation models / Becker-Döring See e.g. work by Aizenman-Back, 79; Ball, Carr, Penrose1990’s; Laurençot 02; Jabin-Niethammer, 03; Fournier-Mischler, 04 ; Dubovski – Stewart ; … • Specificity of this model: particular role of monomers V Coupling v / ui in the “advection” term (replaces coalescence)

11. Discrete model for prion proliferation: Nowak et al, 1998 and Masel et al., 1999 Continuous version: Prüss, Pujo-Menjouet, Webb, Zacher, 2002 Question: is the continuous framework valid ? (Investigated in D, Goudon, Lepoutre, Comm. Math. Sc., 2009)

12. Previous works: Laurençot-Mischler, 2007 (coagulation-fragmentation) Collet, Goudon, Poupaud, Vasseur, 2002 (Lifshitz-Slyozov)A- Rescale the equations Dimensionless parameters appear in the equation: Mass conservation:

13. B- Define a small parameter And mass conservation: Biological interpretation: if and iM typical size of a polymer

14. C- Assumptions on the coefficients Under the following assumption We set

15. C- Assumptions on the coefficients Assumption on n0 : Biological interpretation: x0=0 iff n0 << iM Lemma. Under the previous assumptions + some compactness, we can define continuous functions m, t, b, k and extract a subsequence εn such that

16. Marie DOUMIC Journées Venakides D- Asymptotic result (based on moment estimates) Theorem (D, Goudon, Lepoutre). If moreover We can extract a subsequence εn such that weak solution of :

17. Marie DOUMIC Journées Venakides D- Asymptotic result (based on moment estimates) Theorem (D, Goudon, Lepoutre). If moreover We can extract a subsequence εn such that weak solution of :

18. Marie DOUMIC Journées Venakides E- What to choose for a boundary condition ? Discrete model: no boundary condition. Continuous model: needs one to be well-posed. If: -> obtained rigorously if -> obtained only (by formal division by x0) otherwise We set (in a weak sense):

19. Marie DOUMIC Journées Venakides Some comments • Several different possible continuous equations as limits of the discrete one, according to assumptions on the coefficients: e.g. fragmentation term Can give depolymerization, a boundary condition or a fragmentation kernel Necessity to obtain orders of magnitude • Coalescence has been neglected but may be added

20. Study of the continuous model Constant coefficient case (reduces to 3 coupled ODE) studied by Engl, Prüss, Pujo-Menjouet, Webb, Zacher Motivation for studying the general case: - Silveira et al: 2-peak size distributions of polymers • Biophysical considerations (Radford et al, PNAS 09) Density of polymers “infectious efficiency”: ≈ Polymerization rate Size of particles

21. Continuous model – Well-Posedness A key feature : Mass balance equation: (Simonnett, Walker, JMAP, 2006 & Laurençot, Walker, J. Evol. Equ. 2007): proved For positive polymerization rate • Existence and uniqueness of classical solutions for regular initial condition and bounded coefficients • Existence of weak solutions for L1 initial condition • Stability of the disease-free steady state when it is unique

22. Continuous model – Steady States • Disease-free steady state: • Disease steady state: For x0=0: only possible if

23. Continuous model – Disease Steady State Linked to the following eigenvalue problem: If : the steady state is for If the eigenvalue increases with V : Stability of the disease steady state is possible

24. Marie DOUMIC Journées Venakides Continuous model – Asymptotic behaviour (Calvez, Lenuzza, Mouthon, Perthame et al, BMB 08 & JBD 08) For a polymerization rate possibly vanishing toward zero: • Stability or unstability of the zero steady state under 2 conditions: • Solution for the eigenvalue problem • The eigenvalue increases with V(t) Tool: General Relative Entropy (see Perthame, Michel, Mischler, etc) • Boundedness of the solution • Necessary assumptions on t to obtain an eigenvector with two peaks

25. Shape of the eigenvector (figures from Calvez, Lenuzza et al, 2008) Constant coefficient case Variable polymerization rate Question: what conditions to ensure existence and uniqueness of a solution to the eigenvalue problem ?

26. Study of the eigenvalue problem (D. Gabriel, M3AS, 2010) The polymerization rate possibly vanishes toward zero Rejoin general problems of cell division (see Michel, M3AS, 2006 and Perthame, 2007), and general aggregation/fragmentation models. Many other possible applications

27. Marie DOUMIC Journées Venakides Study of the eigenvalue problem Theorem (D., Gabriel).Under some technical assumptions, there exists a unique solution to the previous eigenvalue problem, and Application. by General Relative Entropy principle (work by Michel, Mischler, Perthame, Ryzhik,…) define then

28. Assumptions for well-posedness of the eigenvalue problem Some necessary assumptions for the previous theorem • For the kernel k: , • To avoid shattering: • Fragmentation and polymerization rates:

29. Marie DOUMIC Journées Venakides Well-posedness for the Eigenvalue problem – sketch of the proof • Truncated & regularized problem -> existence and uniqueness (Krein-Rutman) • A priori estimates (by integration of the equation against proper weights,…) -> weak compactness in L1 (Dunford Pettis) & convergence of the regularized problem

30. Marie DOUMIC Journées Venakides Eigenvalue problem – Fitness dependency on the coefficients (Calvez, D., Gabriel, in progress) Question: how does the eigenvalue depend on the parameters ? Application: • Asymptotic behaviour (see above: monotonous dependency on V is required) • Better understanding of paradoxical biological behaviour (e.g. : what happens if fragmentation increases ? See Knowles et al, Science 09) - Improve “PMCA” protocole

31. Marie DOUMIC Journées Venakides

32. Marie DOUMIC Journées Venakides Sonication phase: increase of β Incubation phase: natural coefficients Some questions: - Is there an optimal finite λ when β increases ? • How to choose an optimal strategy for incubation/sonication times in order to increase the polymer density as fast as possible ?

33. Marie DOUMIC Journées Venakides Eigenvalue problem – Fitness dependency on the coefficients (Calvez, D., Gabriel, in progress) Fragmentation dependence: Proposition. i) If then

34. Marie DOUMIC Journées Venakides Eigenvalue problem – Fitness dependency on the coefficients (Calvez, D., Gabriel, in progress) Fragmentation dependence: Proposition. ii) If then

35. Eigenvalue problem – Fitness dependency on the coefficients Idea: use a self-similar transformation: if (in the spirit of Escobedo, Mischler, Rodriguez Ricard, 04) Define For one has And Proving gives the result.

36. Marie DOUMIC Journées Venakides Eigenvalue problem – Fitness dependency on the coefficients (Calvez, D., Gabriel, in progress) Polymerization dependency: similarly Proposition.

37. Marie DOUMIC Journées Venakides Eigenvalue problem – Numerical simulations (P. Gabriel)

38. Perspectives • Many open questions concerning the eigenvalue problem / asymptotic behaviour • Optimization of the PMCA protocole (Calvez, Gabriel) • Generalize the original model, and justify its use by asymptotic analysis and experiences (in strong interaction with H. Rezaei) • Apply inverse problem techniques to recover the equation parameters from INRA experiments

39. Marie DOUMIC Journées Venakides To be continued…Thanks to French ANR project grant TOPPAZ Theory and Observation of Protein polymerization in Prion and Alzheimer diseases